Gödel, Tarski, Church, and the Liar

'What is a sadist? A sadist is a person who is kind to a masochist.' When I first heard this joke, I smiled, of course, realizing that the brutality of the sadist is, in fact, kindness to a masochist, which is the source of the comic effect being a(n apparent) contradiction resolved. What I did not realize was that I would have had another reason to smile: the kindness of the sadist tortures the masochist. Well, then how should a sadist treat a masochist? Is this really a joke or might it be a serious scientific question? 1 The writer Arthur Koestler in his study, examining the connection between science and art, thinks that it is both ([K] p.95.): Comic discovery is paradox stated – scientific discovery is paradox resolved. I think, he is utterly right. In fact, it will emerge from what we shall do below that the general ideas underlying some of the main mathematical results of this century, those concerning the incompleteness and undecidability of arithmetic and the undefinability of truth within it can, at least partly, be taken as different ways to resolve the archetype of our initial paradoxical question. The paradoxical nature of the famous Gödel's theorem 2 and its less broadly known but equally important and enlightening relatives, the theorems of Tarski and Church, which together constitute the core of the family of the results of modern mathematical logic describing the theoretical limitations of formal reasoning, is inherited from their ancient and not less famous ancestor, one of the most important of all logical paradoxes, that of the Liar. 3 In fact, our aim in this article is just to show that an abstract formal variant of the Liar paradox constitutes a general conceptual schema that, revealing their common logical roots, connects the theorems referred to above and, at the same time, demonstrates that, in a sense, these are the only possible relevant limitation theorems formulated in terms of truth and provability alone that can be considered as different manifestations of the Liar paradox. On the other hand, as we shall illustrate by a simple example, this abstract version of the paradox opens up the possibility to formulate related results concerning notions other than just those of the truth and provability. To obtain the resolution of the paradox in a general, purely formal wording, we shall refor-mulate the infamous statement of the Liar in …